DYNAMIC STATE ESTIMATION OF AN OPERATIONAL STATE OF A GENERATOR IN A POWER SYSTEM

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161. 61. (canceled)
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Abstract
Example embodiments described herein are directed towards dynamic state estimation of an operating state of a generator in a power system. Such estimation is performed for an individual generator in real time with improved accuracy and without the use of Global Position System (GPS) synchronization.
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81 Claims
 161. 61. (canceled)
 62. A method for Dynamic State Estimation, DSE, of an operational state of a generator in a power system, the method comprising:
receiving measured voltage and current analog signals associated with the generator; sampling the received measured signals to voltage and current discrete signals; estimating magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances, respectively, for each variable, using the discrete voltage and current signals as an input in a Discrete Fourier Transform, DFT; calculating a power variable and associated power variance based on the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable; and estimating the dynamic state of the generator using at least a subset of the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable, and the calculated power variable and the associated power variance using a state estimator.  View Dependent Claims (63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79)
 80. An apparatus for Dynamic State Estimation, DSE, of an operational state of a generator in a power system, the apparatus comprising:
a transceiver to receive measured voltage and current analog signals associated with the generator; a processor to sample the received measured signals to voltage and current discrete signals; the processor to estimate magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances, respectively, for each variable, using the discrete voltage and current signals as an input in a Discrete Fourier Transform, DFT; the processor to calculate a power variable and associated power variance based on the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable; and the processor to estimate the dynamic state of the generator using at least a subset of the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable, and the calculated power variable and the associate power variance using a state estimator.
 81. A computer readable medium having executable instructions stored thereon which, when executed by an apparatus for Dynamic State Estimation, DSE, of an operational state of a generator in a power system, cause the apparatus to:
receive measured voltage and current analog signals associated with the generator; sample the received measured signals to voltage and current discrete signals; estimate magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances, respectively, for each variable, using the discrete voltage and current signals as an input in a Discrete Fourier Transform, DFT; calculate a power variable and associated power variance based on the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable; and estimate the dynamic state of the generator using at least a subset of the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable, and the calculated power variable and the associate power variance using a state estimator.
1 Specification
Example embodiments described herein are directed towards dynamic state estimation of an operating state of a generator in a power system. Such estimation is performed for an individual generator in real time with the use of calculated variances, thereby providing improved estimation accuracy. According to some embodiments, the estimation may utilize the calculation of a relative angle of an individual generator thereby providing estimation without the use of Global Position System (GPS) synchronization.
A disturbance in a power system (such as a fault) can initiate spontaneous oscillations in the powerflows in transmission lines. These oscillations grow in magnitude within few seconds if they are undamped or poorly damped. This may lead to loss in synchronism of generators or voltage collapse, ultimately resulting in widescale blackouts. The power blackout of Aug. 10, 1996 in the Western Electricity Coordination Council region is a famous example of blackouts caused by such oscillations.
A generator'"'"'s voltage, current and power are sinusoidal quantities, and since each sinusoid has a magnitude and a phase (which are together known as a phasor), these quantities can either be represented as sine waves, or as phasors. The conversion of sine waves to phasors is done by phasor measurement units (PMUs). In order to estimate the rotor angle any Dynamic State Estimation (DSE) algorithm available in modern power systems typically utilize synchronized measurements obtained using PMUs.
One problem with current state estimation methods is the level of error in the estimations. A further problem with current state estimation is time synchronization is that it has associated noise and synchronizationerrors. Synchronization errors increase the total vector error (TVE) of PMU measurements. As synchronized measurements are used for DSE, these errors can get propagated to the estimated states and deteriorate the overall accuracy and robustness of estimation. It is also not possible to completely eliminate time synchronization as it is inherently required for estimation of rotor angles.
Example embodiments directed herein provide for more accurate estimations as a greater number of inputs and measurements, as well as variances for such inputs and measures, are used in the estimation. Furthermore, some of the example embodiments presented herein provide for a means of decentralized DSE where the need for time synchronization may be eliminated.
Accordingly, the example embodiments presented herein are directed towards an apparatus, computer readable medium and corresponding method for Dynamic State Estimation (DSE) of an operational state of a generator in a power system. The apparatus comprises a transceiver to receive measured voltage and current analog signals associated with the generator. The apparatus further comprises a processor to sample the received measured signals to voltage and current discrete signals. The processor is further to estimate magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances, respectively, for each variable, using the discrete voltage and current signals as an input in a Discrete Fourier Transform (DFT). The processor is also to calculate a power variable and associated power variance based on the estimated magnitude, phase and frequency variables for the voltage and current, as well as associated voltage and current variances for each variable. The processor is further to estimate the dynamic state of the generator using at least a subset of the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable, and the calculated power variable and the associate power variance using a state estimator.
The example embodiments described herein provide a means of DSE with improved accuracy as a greater number of measurements and inputs are used in conjunction with the DFT and state estimator. Furthermore, the example embodiments presented herein improve the accuracy of such estimates with the use of calculated variances.
According to some of the example embodiments, the estimation of the dynamic state of the generator may further comprise estimating a relative angle as a difference between a rotor angle and an estimated voltage phase. The processor may provide such estimation.
According to some of the example embodiments, with the use of the estimation of a relative angle, time synchronization is not needed. Therefore, DSE may be provided in a decentralized manner where the dynamic states may be utilized for decentralized control of a particular generator.
The foregoing will be apparent from the following more particular description of the examples provided herein, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the examples provided herein:
The following sections provides a listing of the mathematical nomenclature, and their corresponding meaning, as used throughout this document:
α difference of rotor angle and stator voltage phase in rad
0 denotes a zero matrix (or vector) of appropriate size
χ denotes a state sigma point
γ denotes a measurement sigma point
g,
h a column vector of the system algebraic functions
I denotes an identity matrix of appropriate size
K the Kalman gain matrix
P denotes a covariance matrix or crosscovariance matrix
u′, v column vectors of pseudoinputs and process noise, respectively
w′, w column vectors of noise in y and u′, respectively
x, y column vectors of states and measurements, respectively
X composite state vector
δ rotor angle in rad
{circumflex over ( )},
λ denotes the λ^{th }component of a DFT
ω, ω_{0 }rotorspeed and its synchronous value in rad/s, respectively
Ψ_{1d }subtransient emfs due to d axis damper coil in p.u.
Ψ_{2q }subtransient emfs due to q axis damper coil in p.u.
σ denotes standard deviation, with σ^{2 }as variance
σ phase of Y'"'"'s fundamental component in rad
θ_{V}, θ_{I }phases of stator voltage and current, respectively, in rad
D rotor damping constant in p.u.
E′_{d }transient emf due to flux in qaxis damper coil in p.u.
E′_{q }transient emf due to field flux linkages in p.u.
E′_{fd }field excitation voltage in p.u.
f frequency of Y'"'"'s fundamental component in Hz
f_{s }sampling frequency for interpolatedDFT method in Hz
f_{V}, f_{0 }frequency of V in p.u. and its base value in Hz, respectively
H generator inertia constant in s
h Hann window function
I, I_{m }analogue stator current and its magnitude, respectively, in p.u.
i, j denote the i^{th }generation unit and √{square root over (−1)}, respectively
I_{d}, I_{q }daxis and qaxis stator currents, respectively, in p.u.
k,
K_{a }AVR gain in p.u.
K_{d1 }the ratio (X″_{d}−X_{l})/(X_{d}′−X_{l})
K_{d2 }the ratio (X_{d}′−X″_{d})/(X_{d}′−X_{l})
K_{q1 }the ratio (X_{q}″−X_{l})/(X_{q}′−X_{l})
K_{q2 }the ratio (X_{q}′−X″_{q})/(X_{q}′−X_{l})
m, n number of states in x and X, respectively, (n=m+2)
N, M total samples for finding DFT and total generation units, respectively
P_{e }active electricalpower output of a machine in p.u.
R_{s }armature resistance in p.u.
t system time in s
T, T_{0 }denote transpose and UKF'"'"'s sampling period (in s), respectively
T_{e}, T_{m0 }electrical and mechanical torques, respectively, in p.u.
T_{r }time constant for the AVR filter in s
T_{d0}′, T_{q0}′ daxis and qaxis transient time constants, respectively, in s
T_{d0}″, T_{q0}″ daxis and qaxis subtransient time constants, respectively, in s
V, V_{m }analogue stator voltage and its magnitude, respectively, in p.u.
V_{d}, V_{q }daxis and qaxis stator voltages, respectively, in p.u.
V_{r}, V_{ref }AVRfilter voltage and AVRreference voltage, respectively, in p.u.
W DFT of Hann window function
X_{d}, X_{q }daxis and qaxis synchronous reactances, respectively, in p.u.
X_{d0}′, X_{q0}′ daxis and qaxis transient reactances, respectively, in p.u.
X_{d0}″, X_{q0}″ daxis and qaxis subtransient reactances, respectively, in p.u.
X_{l }armature leakage reactance in p.u.
Y denotes a sinusoidal signal with harmonics and noise
Y_{m }magnitude of Y'"'"'s fundamental component in p.u.
Z DFT of the product of Y and h
In the following description, for the purposes of explanation and not limitation, specific details are set forth, such as particular components, elements, techniques, etc. in order to provide a thorough understanding of the examples provided herein. However, the examples may be practiced in other manners that depart from these specific details. In other instances, detailed descriptions of wellknown methods and elements are omitted so as not to obscure the description of the examples provided herein.
Example embodiments described herein are directed towards dynamic state estimation of an operating state of a generator in a power system. A disturbance in a power system (such as a fault) can initiate spontaneous oscillations in the powerflows in transmission lines. In order to monitor and control such oscillations and related dynamics which cause instability, the operating state of the system needs to be estimated in realtime, with update rates which are in time scales of ten milliseconds or less (as the time constants associated with such oscillations are not more than ten milliseconds), and this realtime estimation of operating state is known as dynamic state estimation (DSE).
The dynamic states which are estimated and obtained as outputs from DSE algorithms are angles, speeds, voltages and fluxes of the rotors of all the generators in the power system. The inputs which are given to DSE algorithms are some measurable timevarying quantities such as voltage and current of the stator, and some measurable timeinvariant quantities such as resistances, reactances, inertia and other constants for the generator. The constant quantities are measured beforehand, and are used as parameters in DSE algorithms.
A generator'"'"'s voltage, current and power are sinusoidal quantities, and since each sinusoid has a magnitude and a phase (which are together known as a phasor), these quantities can either be represented as sine waves, or as phasors. The conversion of sine waves to phasors is done by phasor measurement units (PMUs). In order to estimate the rotor angle any Dynamic State Estimation (DSE) algorithm available in power system current systems typically utilize synchronized measurements obtained using PMUs.
Example embodiments presented herein provide a means of improved DSE where greater number of inputs and measurements, as well as variances for such inputs and measurements, are utilized in the estimation. Thus, providing improved accuracy for the measurements. According to some of the example embodiments, a means for DSE is provided without the use of time synchronization.
One problem with time synchronization is that it has associated noise and synchronizationerrors. Synchronization errors increase the total vector error (TVE) of PMU measurements. As synchronized measurements are used for DSE, these errors can get propagated to the estimated states and deteriorate the overall accuracy and robustness of estimation. It is also not possible to completely eliminate time synchronization as it is inherently required for estimation of rotor angles.
According to some of the example embodiments, it has been appreciated that although time synchronization is typically used for the estimation of the rotor angle, it is not needed for estimation of other dynamic states, such as rotor speed, rotor voltages and fluxes, as these states are not defined with respect to a common reference angle. Thus, if the dynamic model which is used for estimation can be modified in such a way that rotor angle is replaced with another angle which does not require timesynchronization, then this can minimize the effects of synchronization on accuracy and robustness of estimation.
Some of the example embodiments provide an algorithm for DSE which realizes the above concept. This is done by modifying the estimation model to estimate a relative angle (which does not require synchronization) instead of rotor angle. One such angle is the difference between the rotor angle and the generator terminal voltage phase, also known as the internal angle of the generator. As the rotor angle and the voltage phase have a common reference angle, this reference angle gets cancelled in the difference of the two quantities. Thus, the internal angle, rotor speed, voltages and fluxes can be estimated using the modified estimation model without requiring any synchronized measurements. These dynamic states can then be utilized for decentralized control of the generator. It should be noted that, according to some of the example embodiments, if the estimation of rotor angle is specifically required then it can be indirectly estimated as the sum of the estimated internal angle and the measured terminal voltage phase obtained using PMU.
An example advantage of some of the example embodiments provided herein is to provide a system and method of DSE in which dynamic states are estimated without any time synchronization by incorporating internal angle in estimation model, which in turn ensures robustness of the method to synchronization errors.
The error in phasor measurements considered in several existing methods of DSE is much less than 1% TVE. Such methods of DSE do not consider realistic errors in measurements. The example embodiments presented herein considers and remains accurate for varying levels of errors in measurements—from 0.1% to 10%. Furthermore, none of the currently available methods take into account GPS synchronization errors.
As synchronization is not required for estimation of the states, according to some of the example embodiments, DSE for these states can be performed using the analogue measurements directly acquired from current transformers (CTs) and voltage transformers (VTs). This is particularly beneficial for decentralized control purposes.
According to some of the example embodiments, a dualstage estimation process has been proposed in which a Discrete Fourier transform (DFT) and a state estimator. According to some of the example embodiments, the DFT may be an interpolated DFT and the state estimator may be an unscented Kalman filtering (UKF). According to some of the example embodiments the DFT and the state estimator have been combined as two stages of estimation. The DFT stage dynamically provides estimates of means and variances of the inputs required by the state estimation stage, and this continuous updating of variances may provide noiserobustness of the proposed example embodiments. In existing methods of DSE for power systems, only static estimates of measurement variances are provided to the estimator.
According to some of the example embodiments, analytical expressions have been obtained for the means and variances of the parameter estimates of a sinusoidal signal (which are given as input to the state estimation stage from the DFT stage). Most of these expressions are currently not available in literature.
Rest of the description is organized as follows. First, decoupled equations which may be utilized in conjunction with the example embodiments presented herein are discussed under the subheading ‘Power System Dynamics in a Decoupled Form’. Thereafter, the process for estimation of magnitude, phase and frequency of the analogue signals of terminal voltage and current is discussed under the subheading ‘DFT based Estimation’. An explanation as to how these estimates may be further used for DSE using a state estimator is provided under the subheading ‘State Estimation’. A working example featuring simulations to demonstrate the development of the example embodiments presented herein is provided under the subheading ‘Case Study’. An example node configuration of an apparatus that may be utilized in providing DSE is discussed under the subheading ‘Example Node Configuration’. Finally, an operational flow is provided under the ‘Example Operations’.
Power System Dynamics in a Decoupled Form
A power system comprises a wide variety of elements, including generators, their controllers, transmission lines, transformers, relays and loads. All these elements are electrically coupled to each other, and, therefore, in order to define a power system using dynamic mathematical equations, knowledge of the models, states and parameters of all these constituent elements is useful. Acquiring this knowledge in realtime is not feasible as power systems span wide geographic regions, which are as large as a country, or even a continent. Therefore, according to some of the example embodiments, dynamic equations of the power system in a decoupled form is utilized, so that the realtime estimation of dynamic states can be conducted. According to some of the example embodiments, the estimation may be performed in a decentralized manner.
Such a decoupling of system equations may be achieved if a generator and its controller(s) is considered as a decentralized unit, and the stator terminal voltage magnitude, Vm, and its phase, θV, are treated as ‘inputs’ in the dynamic equations, instead of considering them as algebraic quantities or measurements. This concept is referred to herein as ‘pseudoinputs’ for decoupling the equations.
In order to estimate the internal angle, which is the difference between the rotor angle and the voltage phase, instead of estimating the rotor angle, the decoupled equations and the pseudoinputs for a generator are altered. The altered decoupled equations are given by equations (1)(11), derived using the subtransient model of machines with four rotor coils in each machine, known as IEEE Model 2.2 as provided in “IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses,” IEEE Std 11102002 (Revision of IEEE Std 11101991), pp. 172, 2003. In these equations, the altered pseudoinputs are Vm and voltage frequency, fV, and i refers to the system'"'"'s ith machine or generator, 1≤i≤M. Slow dynamics of the speedgovernor have been ignored in this model (although they can also be added, if required). Also, model of a static automatic voltage regulator (AVR) is included with the model of each machine.
The above equations may be written in the following composite statespace form which may be used for DSE (here pseudoinputs are denoted by u′_{i}, and the process noise and the noise in pseudo inputs have been included, and denoted by v^{i }and ω′^{i}, respectively).
{dot over (x)}^{i}=g′^{i}(x^{i},u′^{i},w′^{i})+v^{i};u′^{i}−w′^{i}=[V_{m}^{i }f_{V}^{i}]^{T }
x^{i}=[α^{i}ω^{i }E′_{d}^{i }E′_{q}^{i}Ψ_{1d}^{i}Ψ_{2q}^{i }V_{r}^{i}]^{T} (12)
DFT Based Estimation
The example embodiments described herein make use of an interpolated DFT for estimating the parameters of a sinusoidal signal. It should be appreciated that other forms of DFTs may be utilized in the estimation of the parameters. The use of an interpolated DFT based estimation has the example advantage of being both fast and accurate enough for realtime control applications in power systems. According to some of the example embodiments, the DFT may be used for finding the estimates of frequency, magnitude and phase of the fundamental components of measurements obtained from CTs and PTs.
The fundamental component of a sinusoidal signal can be extracted by multiplying the signal with a suitable window function which eliminates other harmonics and higher frequency components in the signal, followed by finding its DFT. One such function is, for example, a Hanning window function given by hk=sin 2(πk/N), and if this function is multiplied with N samples of an analogue signal Y(t) sampled at a frequency fs, then DFT of the product is given by Z(λ) as follows.
where, Ym, θ and f are magnitude, phase and frequency of Y'"'"'s fundamental component, respectively; λ∈{0, 1, . . . , N−1}; and W(λ) is the following DFT of Hanning window function.
A concept in interpolatedDFT based estimation is to approximate W(λ) with the following expression, provided that N>>1 and λ<<N.
By substituting equation (15) in equation (13), Z(λ) can be expressed as follows for N>>1 and λ<<N.
where Ŷ_{m}, {circumflex over (θ)} and {circumflex over (f)} denote the estimates of Y_{m}, θ and f, respectively. As equation (16) has three unknowns (which are Ŷ_{m}, {circumflex over (θ)} and {circumflex over (f)}), three distinct equations are required to estimate these unknowns. This may be done by choosing any three distinct values of λ in equation (16) (say λ=1, λ=2 and λ=3). The obtained values of Ŷ_{m}, {circumflex over (θ)} and {circumflex over (f)} will have associated estimation errors which will depend on N and on the values of λ which are used for generating the three distinct equations. More precisely, these estimation errors are inversely proportional to N^{4}, and, hence, N should be as large as practically feasible. In the example embodiments described herein N is taken to be in the order of 10^{3}, as this is the highest order for N for which interpolatedDFT can run on a stateoftheart DSP processor without overloading it, however it should be appreciated such values are presented herein merely as an example. Overloading refers to overall processor usage of above 95%. Also, for a given N, the estimation errors are minimized if the choices for λ are taken as λ=0, λ=1 and λ=2, provided that
otherwise, for
the errors are minimized if the choices are λ=1, λ=2 and λ=3. The value of
should not be greater than 3 as then the delay in obtaining the estimated values becomes too large (that is, more than two cycles, or more than 0.04 s for a 50 Hz power system), and at the same time it should not be too small as then the accuracy of estimation is diminished. According to some of the example embodiments, an intermediate value of
has been taken and, hence, the former choices of λ=0, λ=1 and λ=2 are applicable.
is an unknown quantity as f needs to be estimated. But because of power system operational requirements, f should remain within 5% of the base system frequency, f_{0 }(which is usually 50 Hz or 60 Hz), and, hence, if N and f_{s }are chosen such that
The 3 equations which are obtained by putting λ=0, λ=1 and λ=2 in equation (16) can be written in matrix form as follows.
Equation (17) implies that the product of a squarematrix and a column vector is equal to a zero vector, when both the matrix and the vector have nonzero elements. This occurs if the columns of the matrix are linearly dependent, that is, the determinant of the matrix is zero, given as follows.
Simplification of the above determinant gives {circumflex over (f)} as follows.
{circumflex over (θ)} may be obtained by substituting the above value of {circumflex over (f)} back into equation (16) and eliminating Ŷ_{m}. To do this, the equation which is obtained by putting λ=0 in equation (16) is divided by the equation obtained by putting λ=1 in equation (16), which comes as follows.
Solving for e^{j{circumflex over (θ)} }using equation (20) gives the following expression.
Using equations (21) and (16) (with λ=0), Ŷ_{m }comes as follows.
where B, C, and e^{j{circumflex over (θ)} }are given by equations (20)(21).
It should be noted that Ŷ_{m}, {circumflex over (θ)} and {circumflex over (f)} are real quantities, but they are obtained as functions of complex quantities (given in the right hand sides (RHSs) of equations (19), (21) and (22), respectively). Hence, these quantities will have negligible but finite imaginary parts associated with them because of finite computational accuracy of any computational device. Thus, during implementation, the imaginary parts should be ignored and only the real parts of RHSs should be assigned to Ŷ_{m}, {circumflex over (θ)} and {circumflex over (f)}. Also, as Ŷ_{m }and {circumflex over (f)} are strictly positive, absolute values of real parts of respective RHSs should be assigned to them.
The variance of the above estimate of {circumflex over (f)} in equation (19) is approximately twice the minimum possible variance which is theoretically achievable using any unbiased estimator (known as CramerRao bound (CRB)). CRB for frequency estimation of a sinusoidal signal and is given by CRB({circumflex over (f)}) (in Hz^{2}) as follows.
where σ_{Y}^{2 }is the variance of noise in Y (in p.u.). CRBs for Ŷ_{m }and {circumflex over (θ)} are given by CRB(Ŷ_{m}) (in p.u.) and CRB({circumflex over (θ)}) (in rad^{2}), respectively, as follows.
The variances of Ŷ_{m }and {circumflex over (θ)} are found to be approximately two and six times the above CRBs in equation (24), respectively; and hence, the estimated variances of {circumflex over (f)}, {circumflex over (θ)} and Ŷ_{m }are given by {circumflex over (σ)}_{f}^{2 }(in p.u.), {circumflex over (σ)}_{Y}_{m}^{2 }(in p.u.) and {circumflex over (σ)}_{θ}^{2 }(in rad^{2}), respectively, as follows.
where CRB({circumflex over (f)}), CRB(Ŷ_{m}) and CRB({circumflex over (θ)}) are given by equations (23)(24). Estimates of means and variances obtained above are given as inputs to the state estimation stage (e.g., UKF), as detailed in the next section.
An example advantage of obtaining the analytical expressions for {circumflex over (σ)}_{f}^{2}, {circumflex over (σ)}_{Y}_{m}^{2}, and {circumflex over (σ)}_{θ}^{2 }in equations (23)(25) is that these variances may be continuously updated and provided to the dynamic estimator (which is the state estimation stage, for example, the UKF stage) along with {circumflex over (f)}, {circumflex over (θ)} and Ŷ_{m}, thereby improving the accuracy of dynamic state estimation.
State Estimation
State estimation will be described herein using an UKF as an example. However, it should be appreciated that other forms of state estimation, or types of Kalman Filters, may be employed. UKF is a nonlinear method for obtaining dynamic state estimates of a system. It employs the idea that performing DSE is easier if the distribution of state estimates is transformed, than if the system model itself is transformed through linearization. System linearization requires computation of Jacobian matrices and is a mathematically challenging task for a high order power system model, especially if it needs to be done at every iteration. Since linearization is not required in UKF, and, moreover, it has higher accuracy and similar computational speeds as that of linear methods of DSE, UKF has been used for performing DSE according to some of the example embodiments, however, other Kalman Filters or forms of state estimation may also be employed. UKF is a discrete method and, hence, the system given by (12) may be discretized before UKF may be applied to it. Discretizing (12) at a sampling period T_{0}, by approximating {dot over (x)}^{i }with (x^{ik}−x^{ik})/T_{0}, gives the following equation (where k and
x^{ik}=x^{ik}+T_{0}ǵ^{l}(x^{ik},ú^{ik},{acute over (w)}^{ik})+v^{ik}⇒x^{ik}=g^{i}(x^{ik},u^{ik},{acute over (w)}^{ik})+v^{ik} (26)
In the above model, {circumflex over (V)}_{m}^{ik }and {circumflex over (f)}_{V}^{ik }(found using the DFT method) are used in the pseudoinput vector ú^{ik }as follows.
ú^{ik}=[{circumflex over (V)}_{m}^{ik}{circumflex over (f)}_{V}^{ik}]^{T}=[V_{m}^{ik }f_{V}^{ik}]^{T}+{acute over (w)}^{ik} (27)
UKF also utilizes a measurement model besides the above process model. The estimates of active power, P_{e}^{ik }(defined by equation (10)), and stator current magnitude, I_{m}^{ik }(defined by equation (11)), which are obtained using the DFT method are used as measurements for UKF. After incorporating the measurement noise, w^{ik}, the measurement model is given as follows.
ú^{ik }and y^{ik }are estimated quantities and have finite variances which may be included in the process and measurement models, respectively. This may be done by including {acute over (w)}^{ik }and w^{ik }in the models as the following zeromean noises.
where P_{{acute over (w)}}^{ik }and P_{w}^{ik }denote the covariance matrices of {acute over (w)}^{ik }and w^{ik}, respectively. In order to find the estimates and the variances in equations (27)(30), the stator voltage, V^{i}(t), and stator current, I^{i}(t), measured using VT and CT, respectively, are processed using the DFT method. Thus, {circumflex over (V)}_{m}^{ik}, {circumflex over (f)}_{V}^{ik}, {circumflex over (θ)}_{V}^{ik}, {circumflex over (σ)}_{V}_{m}_{ik}^{2}, {circumflex over (σ)}_{f}_{V}_{ik}^{2 }and {circumflex over (σ)}_{θ}_{V}_{ik}^{2 }are obtained by putting Y(t)=V^{i}(t) in equations (13)(22) and updating these estimates and variances for every k^{th }sample. Similarly, Î_{m}^{ik}, {circumflex over (f)}_{I}^{ik}, {circumflex over (θ)}_{I}^{ik}, {circumflex over (σ)}_{I}_{m}_{ik}^{2}, {circumflex over (σ)}_{f}_{I}_{ik}^{2 }and {circumflex over (σ)}_{θ}_{I}_{ik}^{2 }are obtained by putting Y(t)=I^{i}(t). As {circumflex over (P)}_{e}^{ik}={circumflex over (V)}_{m}^{ik}Î_{m}^{ik }cos({circumflex over (θ)}_{V}^{ik}−{circumflex over (θ)}_{I}^{ik}) (from equation (10)) and the means values and variances of V_{m}^{ik}, I_{m}^{ik}, θ_{V}^{ik}, and θ_{I}^{ik }are known, the mean value of P_{e}^{ik }(denoted as {circumflex over (P)}_{e}^{ik}) and its estimated variance (denoted as σ_{P}_{e}_{ik}^{2 }can be represented in terms of these known quantities, and have been obtained as follows (here it should be noted that by definition θ_{V}^{ik }and θ_{I}^{ik }lie in the interval (−π/2, π/2], hence, they should be ‘unwrapped’ by adding or subtracting suitable multiples of π to them, in order to find cos({circumflex over (θ)}_{V}^{ik}−{circumflex over (θ)}_{I}^{ik}).
Thus, the four quantities which are utilized by the state estimation stage, for example the UKF stage, from the DFT stage are ú^{i}, y^{i}, P_{{acute over (w)}}^{ik}, and P_{w}^{ik}, given by equations (27)(30). These quantities should be updated ever T_{0 }s, as this is the sampling period of the UKF stage. Also, in equation (26), both x^{ik }and {acute over (w)}^{ik }are unknown quantities and may be combined together as a composite state vector X^{ik }with a composite covariance matrix P_{X}^{ik }defined as follows.
Here P_{X}^{ik }is the covariance matrix of x^{ik}, and P_{x{acute over (w)}}^{ik }is the crosscovariance matrix of x^{ik }and {acute over (w)}^{ik}. With the above definition, the model in equations (26)(28) is redefined as follows.
X^{ik}=g^{i}(X^{ik},ú^{ik})+v^{ik};y^{ik}=h^{i}(X^{ik},ú^{ik})+w^{ik} (33)
With equation (33) as a model and x^{i0 }as a steady state estimate of x^{ik }and with the knowledge of g^{i}, h^{i}, ú^{i}, y^{i}, P_{{acute over (w)}}^{ik}, P_{w}^{ik }and the process noise covariance matrix, P_{v}^{ik}, the filtering equations of the state estimator, for example, the UKF, for the k^{th }iteration of the i^{th }unit are given as follows.
 if (k==1) then initialize {circumflex over (x)}^{ik}=x^{i0}, ŵ′^{ik}=0_{2X1}, P_{x}^{ik}=P_{v}^{i0}, P_{x}^{ik}=0_{m}_{i}_{x2}, P_{w}^{ik}=P_{{acute over (w)}}^{i0 }in equation (32) to get P_{X}^{ik} & {circumflex over (X)}^{ik}.
else reinitialize ŵ′^{ik }and P_{{acute over (w)}}^{ik}, in equation (32) according to equation (29), leaving the rest of the elements in {circumflex over (X)}^{ik }and P_{X}^{ik }unchanged.
K^{ik}=P_{Xy}^{ik−}(P_{y}^{ik−})^{−1};{circumflex over (X)}^{ik}={circumflex over (X)}^{ik−}+K^{ik}(y^{ik}−ŷ^{ik−})
P_{X}^{ik}=P_{X}^{ik}−K^{ik}[P_{Xy}^{ik−}]^{T }
output {circumflex over (X)}^{ik }and P_{X}^{ik}, k←(k+1), go to Operation 1.
Case Study
According to some of the example embodiments, the robust dynamic state estimator (as discussed under the subheadings ‘Power System Dynamics in a Decoupled Form’ and ‘State Estimation’) runs at the location of each generation unit, and provides dynamic state estimates for the unit. The measurements which are provided to the estimator are V (t) and I(t), and are generated by adding noise to the simulated analogue values of terminal voltage and current of the unit. As explained under the subheading ‘DFT based Estimation’N, f_{0 }and f_{s }are taken as 1200, 50 Hz and 40000 Hz, respectively. The sampling period of state estimation stage (e.g., UKF stage), T_{0}, is taken as 0.01 s, and thus, the estimates obtained from the DFT stage are also updated every 0.01 s. Also, P_{v}^{ik }is found. For comparison with the proposed estimator, another UKF based dynamic state estimator which uses PMU measurements also runs at each unit'"'"'s location and is termed as DSEwithPMU. Estimate of the internal angle in case of DSEwithPMU is obtained by subtracting the measurement of terminal voltage phase from the estimate of rotor angle.
The measurement error for the robust DSE method is the percentage error in the analogue signals of V(t) and I(t), while the measurement error for DSEwithPMU method is the total vector error (TVE) in the phasor measurements of terminal voltage and current. As the measurement errors for the two estimators are of two different kinds, these methods may not be directly compared for the same noise levels. Nevertheless, performance of the two methods for standard measurement errors can be compared, as specified by IEEE. As mentioned in these standards, the measurement error in CTs/VTs should be less than 3%, while the standard error for PMUs is 1% TVE. Hence, in the base case for comparison, the measurement error for robust DSE is taken as 3%, while for the DSEwithPMU method, it is taken as 1% TVE. The system starts from a steady state in the simulation. Then at t=1 s, a disturbance is created by a threephase fault at bus 54 and is cleared after 0.18 s by opening of one of the tielines between buses 5354. The simulated states, along with their estimated values for the base case for one of the units (the 13^{th }unit), have been plotted in
It can be seen in
Robustness of the example embodiments has been tested against varying noise levels in measurements.
Computational feasibility of the example embodiments may be inferred from the fact that the entire simulation, including simulation of the power system, with two estimators at each machine, runs in realtime on MATLABSimulink running on Windows 7 on a personal computer with Intel Core 2 Duo, 2.0 GHz CPU and 2 GB RAM. The expression ‘realtime’ here means that 1 second of the simulation takes less than 1 second of processing time. Also, the total execution time for all the operations for the proposed method for one time step (that is for one iteration) is 0.44 millisecond. Thus, the method can be easily implemented using current technologies as the update rate required by the proposed method is 10 milliseconds.
The apparatus may be further configured to calculate a power variable (P_{e}), and corresponding power variance, based on the estimated magnitude, phase and frequency variables of the voltage (V_{m}, θ_{V}, and f_{V}) and current (I_{m}, θ_{V}, and f_{I}), as well as associated voltage and current variances, respectively, for each variable.
Thereafter, the dynamic state (DSE) of the generator may be estimated using at least a subset of the estimated magnitude, phase and frequency variables of the voltage (V_{m}, θ_{V}, and f_{V}) and current (I_{m}, θ_{V}, and f_{I}), as well as associated voltage and current variances, respectively, for each variable, as well as the calculated power variable (P_{e}), and corresponding power variance.
In
The chosen subset may thereafter be input in to the state estimator, for example a UKF. The portions of the subset may be provided to the state estimator as differential equations and portions of the subset may be provided to the state estimator as an algebraic equation. The output of the state estimator is the DSE of the operational state of a generator in the power system. According to some of the example embodiments, as the DFT and the state estimator make use of estimated and calculated variances, a more accurate DSE may be provided.
The apparatus 10 of
The apparatus 10 may further comprise at least one memory 16 that may be in communication with the transceiver and the processor. The memory 16 may store received or transmitted data, processed data, and/or executable program instructions. The memory may be any suitable type of machine readable medium and may be of a volatile and/or nonvolatile type.
Operation 20
The apparatus 10 is to receive 20 measured voltage and current analog signals associated with the generator. The transceiver 12 is to receive measured voltage and current analog signals associated with the generator.
Operation 22
The apparatus 10 is to sample 22 the received measurement signals to voltage and current discrete signals. The processor 14 is to sample the received measurement signals to voltage and current discrete signals.
According to some of the example embodiments, the sampling 22 further comprises multiplying 24 the voltage and current discrete signals with a window function. The processor may multiple the voltage and current discrete signals with the window function.
According to some of the example embodiments, the multiplication with the window function may further comprise a multiplication according to
Example operation 24 is further described under at least the subheading ‘DFT based Estimation’.
Operation 26
The apparatus is further to estimate 26 magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances, respectively, for each variable, using the discrete voltage and current signals as an input in a DFT. The processor 14 is to estimate the magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances, respectively, for each variable, using the discrete voltage and current signals as an input in a DFT.
According to some of the example embodiments, the estimation 26 may be provided according to
An example advantage of operation 26 is that with the calculation of the variances for respective variables, DSE with greater accuracy may be achieved. Operation 26 is further described under at least the subheading ‘DFT based Estimation’.
Operation 28
The apparatus is further to calculate a power variable and associated power variance based on the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable. The processor 14 is to calculate the power variable and associated power variance based on the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable.
According to some of the example embodiments, the calculated power may be a real power or a reactive power. According to some of the example embodiments, the calculation of the power variable and the associate variance further comprises a calculation according to
Operation 28 is further described under at least the subheading ‘State Estimation’.
According to some of the example embodiments, the DFT is an interpolated DFT. Example operation 30 is further described under at least the subheading ‘DFT based Estimation’.
Operation 32
The apparatus 10 is further to estimate the dynamic state of the generator using at least a subset of the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable, and the calculated power variable and the associated power variance using a state estimator. The processor 14 is to estimate the dynamic state of the generator using at least the subset of the estimated magnitude, phase and frequency variables of the voltage and current, as well as associated voltage and current variances for each variable, and the calculated power variable and the associated power variance using the state estimator.
According to some of the example embodiments, the estimating 32 may further comprise estimating the DSE according to
According to some of the example embodiments, the estimating 32 of the dynamic state is based on, at least in part, estimating 34 a relative angle as a difference between a rotor angle and the estimated voltage phase. The processor 14 may estimate the relative angle as the difference between the rotor angle and the estimated voltage phase.
According to some of the example embodiments, the estimating of the relative angle further comprises estimating the relative angle according to Δ{dot over (α)}^{i}=(ω^{i}−f_{V}^{i}). An example advantage of example operation 34 is that with the use of a relative angle, time synchronization may be avoided. Example operation 34 is further described under at least the subheading ‘Power System Dynamics in a Decoupled Form’.
According to some of the example embodiments, the estimating 32 is based on a subset of the estimated magnitude of the voltage, the estimated frequency of the voltage and the estimated magnitude of the current. The processor 14 may base the estimation of the dynamic states on a subset comprising the estimated magnitude of the voltage, the estimated frequency of the voltage and the estimated magnitude of the current.
According to the embodiments in which the subset is as described in example operation 36, the estimated magnitude of the voltage and the estimated frequency of the voltage and associated voltage variances are inputs to the state estimator, and are utilized as pseudoinputs, and are provided in differential equations of the state estimator, and the estimated magnitude of the current and the calculated power and associated current and power variances are provided as an algebraic equation and are given as measurement inputs to the state estimator.
According to such example embodiments, the differential equations are represented as x^{ik}=x^{ik}+T_{0}ǵ^{l}(x^{ik}, ú^{ik}, {acute over (w)}^{ik})+v^{ik}⇒x^{ik}=g^{i}(x^{ik}, u^{ik}, {acute over (w)}^{ik})+v^{ik}, where x is a column vector of a state,
Example operation 38 is further described under at least subheading ‘State Estimation’.
According to some of the example embodiments, the estimating 32 is based on a subset of the estimated magnitude of the current, the estimated frequency of the current and the estimated magnitude of the voltage. The processor 14 is to estimate the dynamic state based on the subset of the estimated magnitude of the current, the estimated frequency of the current and the estimated magnitude of the voltage.
According to the embodiments in which the subset is as described in example operation 40, the estimated magnitude of the current and the estimated frequency of the current and associated current variances are inputs to the state estimator and are used in differential equations of the state estimator, and the estimated magnitude of the voltage and the calculated power and associated voltage and power variances are represented as an algebraic equation and is given as a measurement input to the state estimator.
It should further be appreciated that other such subsets may be utilized. For example, according to some of the example embodiments, the subset of the estimated magnitude, phase and frequency variables of the voltage and current is the estimated magnitude of the voltage, the estimated magnitude of the current and the estimated frequency of the voltage. According to such embodiments, the estimated magnitude of the voltage and the estimated magnitude of the current and associated voltage and current variances are inputs to the state estimator and are used in differential equations of the state estimator, and the estimated frequency of the voltage and the calculated power and associated voltage and power variances are represented as an algebraic equation and is given as a measurement input to the state estimator, wherein the calculated power is reactive power.
According to some of the example embodiments, the subset of the estimated magnitude, phase and frequency variables of the voltage and current is the estimated magnitude of the current, the estimated frequency of the current and the estimated magnitude of the voltage. According to such example embodiments, the estimated magnitude of the current and the estimated frequency of the current and associated current variances are inputs to the state estimator and are used in differential equations of the state estimator, and the estimated magnitude of the voltage and the calculated power and associated voltage and power variances are represented as an algebraic equation and is given as a measurement input to the state estimator. Example operation 42 is further described under at least subheading ‘State Estimation’.
According to some of the example embodiments, the estimating 32 further comprises estimating the DSE with the use of a UKF as the state estimator. The processor 14 may estimate the DSE with the use of the UKF as the state estimator.
The various example embodiments described herein are described in the general context of method steps or processes, which may be implemented in one aspect by a computer program product, embodied in a computerreadable medium, comprising computerexecutable instructions, such as program code, executed by computers in networked environments. A computerreadable medium may comprise removable and nonremovable storage devices comprising, but not limited to, Read Only Memory (ROM), Random Access Memory (RAM), compact discs (CDs), digital versatile discs (DVD), etc. Generally, program modules may comprise routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Computerexecutable instructions, associated data structures, and program modules represent examples of corresponding acts for implementing the functions described in such steps or processes.
Throughout the description and claims of this specification, the words “comprise” and “contain” and variations of them mean “including but not limited to”, and they are not intended to (and do not) exclude other moieties, additives, components, integers or steps. Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.
Features, integers, characteristics or groups described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith. All of the features disclosed in this specification (including any accompanying claims, abstract and drawings), and/or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and/or steps are mutually exclusive. The example embodiments not restricted to the details of any foregoing embodiments. The example embodiments extend to any novel one, or any novel combination, of the features disclosed in this specification (including any accompanying claims, abstract and drawings), or to any novel one, or any novel combination, of the steps of any method or process so disclosed.
The reader'"'"'s attention is directed to all papers and documents which are filed concurrently with or previous to this specification in connection with this application and which are open to public inspection with this specification, and the contents of all such papers and documents are incorporated herein by reference.
In the drawings and specification, there have been disclosed exemplary embodiments. However, many variations and modifications may be made to these embodiments. Accordingly, although specific terms are employed, they are used in a generic and descriptive sense only and not for purposes of limitation, the scope of the embodiments being defined by the following claims.